The ignition delay of a dual fuel system has been numerically investigated by adopting a constant volume chamber as a model problem simulating diesel engine relevant conditions. A detailed chemical kinetic mechanism, consisting of 28 species and 135 elementary reactions, of dimethyl ether (DME) with methane ($CH_{4}$) sub-mechanism has been used in conjunction with the multi-dimensional reactive flow KIVA-3V code to simulate the autoignition process. The start of ignition was defined as the moment when the maximum temperature in the combustion vessel reached to 1900 K with which a best agreement with existing experiment was achieved. Ignition delays of liquid DME injected into air at various high pressures and temperatures compared well with the existing experimental results in a combustion bomb. When a small quantity of liquid DME was injected into premixtures of $CH_{4}$/air, the ignition delay times of the dual fuel system are longer than that observed with DME only, especially at higher initial temperatures. The variation in the ignition delay between DME only and dual fuel case tend to be constant for lower initial temperatures. It was also found that the predicted values of the ignition delay in dual fuel operation are dependent on the concentration of the gaseous $CH_{4}$ in the chamber charge and less dependent on the injected mass of DME. Temperature and equivalence ratio contours of the combustion process showed that the ignition commonly starts in the boundary at which near stoichiometric mixtures could exists. Parametric studies are also conducted to show the effect of additive such as hydrogen peroxide in the ignition delay. Apart from accurate predictions of ignition delay, the coupling between multi-dimensional flow and multi-step chemistry is essential to reveal detailed features of the ignition process.